"Does Infinity Really Exist?"

Sorry for the late notice, but this upcoming live-streamed discussion of infinity (part of the World Science Festival) looks like it should be great (8pm.-9:30pm ET TONIGHT!!) if you've got no set plans for the evening:

http://www.scientificamerican.com/article.cfm?id=does-infinity-really-exist

 Moderated by Keith Devlin, with panelists Raphael Bousso, Philip Clayton, Steven Strogatz and W. Hugh Woodin. ...Like I said, oughta be great!


ADDENDUM: if you missed it, here it is:



It's fully two hours long and I found the first half-hour a bit dull/mundane (if you have any background at all with infinity), but for me it got more interesting with the entry of Hugh Woodin at around the 33-minute point and the physicist, Raphael Bousso (really enjoyed the chance to hear Woodin, who I'd not heard before and who is well-known for new work on the Continuum Hypothesis).

Corporate Math...

Math teacher Patrick Honner recently took on a Prudential commercial to point out the "subtle ways" that corporate interests use 'mathematics to manipulate opinions' (...he ended up taking on some commenters as well):

 http://mrhonner.com/2013/05/28/how-old-is-the-oldest-person-you-know/ 

Of course we're awash in the business use of math/statistics to influence our behavior.

Anyway, it all reminded me ever-so-slightly of a comical clip I used here once before (...all in the name of comedy folks):



ADDENDUM: well, just discovered that the same Don McMillan has another clip even more pertinent to "marketing" statistics:

Remembering... Gardner

"It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.
"I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view." -- John Conway

Not to take anything away from our Veterans, but this is a math blog, and I'll use the opportunity of Memorial Day to once again remember Martin Gardner, whose death just over 3 years ago inspired me to start this endeavor (with no idea it would still be up-and-running 3 years later!!).

The above quote from John Conway, one of the most creative, productive mathematicians around, is taken from a book review Gardner wrote for a 2009 volume by Mariana Cook, covering 92 mathematicians, entitled "Mathematicians: An Outer View of the Inner World."
It's not a book I've personally seen, but read Gardner's review here:

http://www.newcriterion.com/articles.cfm/Still-four-4349

Gardner (a vocal math Platonist) uses the review to go off on the topic of Platonist vs. non-Platonist viewpoints, writing at one point, "I suspect that almost every mathematician in the book is a Platonic realist, one who believes that mathematical theorems are forever true in all possible worlds and are independent of human culture," before continuing on to offer the above quote from Conway. If anything, the Platonism divide has only deepened since 2009, with brilliant adherents on both sides -- I can't help but think some of it is little more than muddy semantics, while also recognizing that there does exist a core of real (and perhaps non-resolvable) disagreement.


On a side note, I see that Martin Gardner's forthcoming autobiography "Undiluted Hocus-Pocus: The Autobiography of Martin Gardner" is already listed on Amazon, and I suspect will include more of his Platonist evangelism (available in September):

http://www.amazon.com/Undiluted-Hocus-Pocus-Autobiography-Martin-Gardner/dp/0691159912

….I almost wish I didn't know it was on the way... because the 4-month wait will now be excruciating!! :-/

Lastly, if in the mood for some more memories of Martin see here:

http://www.ams.org/notices/201103/rtx110300418p.pdf

also, this great 2005 AMS interview with Martin:

http://www.ams.org/notices/200506/fea-gardner.pdf 

Everybody Loves Raymond!

Just came to my attention that today is Raymond Smullyan's 94th birthday. THAT is an occasion worth noting!
As I've written before, Smullyan, like Martin Gardner, is an American gem!

An hour-long 2004 movie about him, aptly titled "This Film Needs No Title: A Portrait of Raymond Smullyan," (after a book of his entitled, "This Book Needs No Title"), is available (with registration) here:

http://tinyurl.com/owxc53k

A review of his interesting 1977 NON-mathematical volume, "The Tao Is Silent" here:

http://ramblingtaoist.blogspot.com/2009/08/smullyan-on-tao.html
(of course he's written a plethora of math/logic/puzzle books as well, but the above offering is a bit different)

For the umpteenth time (apologies to long-time readers here) I'll link back to a post on one of my very favorite Smullyan puzzles/paradoxes which I re-cite every year:

http://math-frolic.blogspot.com/2011/01/seemingly-impossible-task-that-isnt.html

And finally a fun, self-referential paradox presented by Richard Elwes which I believe (like one of Elwes' commenters) also originated, in some form, from Smullyan:

http://richardelwes.co.uk/2011/06/06/an-idiotic-paradox/

HAPPY BIRTHDAY DR. SMULLYAN! (and may the Tao be with you!)

Woit, Weinstein, Lisi... Oh My!

For the physics crowd....

Eric Weinstein via Wikipedia

When Peter Woit writes about mathematician/economist Eric Weinstein and surfer-physicist Garrett Lisi in the same post, well, I have to take note of it!:

http://www.math.columbia.edu/~woit/wordpress/?p=5927

Weinstein is apparently giving a lecture at Oxford today (invited by Marcus du Sautoy) on some mathematical physics ideas (called "Geometric Unity") he has been working on for awhile (but not shared much about), which Woit analogizes to the outside-the-box thinking of surfer/physicist Garrett Lisi. Further, Peter writes:

"Both he [Weinstein] and Garrett are pursuing what seems to me one of the deepest questions around: what is the relationship between the SU(3)xSU(2)xU(1) geometry of the Standard Model, and the 4d pseudo-Riemannian geometry of space-time and general relativity? Garrett was trying to understand this in terms of E(8) symmetry, and I’m looking forward to seeing what Eric’s ideas about this are."

A bit above my pay grade, but still interesting-sounding stuff! Maybe in a day or two we'll hear more about it.

Until then, you can also read this long Guardian piece on Weinstein's work, which Peter linked to as well:

http://www.guardian.co.uk/science/blog/2013/may/23/roll-over-einstein-meet-weinstein

Over-the-hill… or, Not!

"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. ... I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself."

That's what esteemed British mathematician G. H. Hardy famously wrote back in 1940.

According to at least one source, Yitang Zhang, conveyor of the recent much-heralded 'bounded gap' proof for prime numbers, is in his 50's.

A number of general press pieces continue to follow the Zhang story, pretty much re-hashing the same information already released. RJ Lipton has added a little more added thought (if you're prepared for some deeper math) at his "Gödel's Last Letter…" blog here:

http://rjlipton.wordpress.com/2013/05/21/twin-primes-are-useful/

He includes several links as well, including one that led me to an old MathOverflow thread on "Major mathematical advances past fifty" which begins with the above quote... before offering an array of counter-examples:

http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty-closed

…. a possibly interesting/encouraging read for some of us, of a certain age. ;-)

Hans Rosling... "Zen Master"

At his Discover blog, Keith Kloor has put up a post on Hans Rosling, Swedish doctor/statistician who has gained a wide following via the TEDTalk series, and who Keith labels "the Zen master of statistics":

http://blogs.discovermagazine.com/collideascape/2013/05/20/the-zen-master-of-statistics/
(includes several video clips of Hans in action)

Conspiracy Theorists, Sleuths, Sherlock Holmes' Fans Come Hither!

Well, this is interesting… in a weird, bizarre, off-the-wall kind of way....

Ted Nelson, an established "computer visionary," claims in a 13-minute video to have unmasked the identity of the creator of Bitcoin, the digital currency that has been the focus of much financial news in recent times (but whose actual inventor has never been divulged).      

I've followed Bitcoin news for awhile now, but I don't even pretend to understand the mechanics of how the currency system works or even came into being... I've seen predictions ranging from it being the certain and eventual future of worldwide currency, to it being a fad that will crash-and-burn taking down a great many of its speculative proponents in the process.

Anyway, the alias long-used by Bitcoin's creator is "Satoshi Nakamoto," but Nelson believes he has deduced that the true instigator of Bitcoin is… ready for the drrr-r-r-r-rumroll… none other than mathematician Shinichi Mochizuki, recent elusive claimant of a proof for the ABC conjecture!!

Definitely fun to watch Nelson lay out his entertaining (though I think quite flimsy), case here:



Forbes article on the matter here:

http://tinyurl.com/a4xdv8t

I wouldn't put much stock in Nelson's theory, but let's face it, we all love a mystery, and playing detective....

Math For the Masses from NY Times

Anthologies of popular math writing don't come along all that often so I was pleased to see that the NY Times has put out "The New York Times Book of Mathematics," a compendium of non-technical articles that have appeared in the newspaper from 1892 to 2010. I've only leafed through it briefly in a bookstore, but with over 100 selections edited by Gina Kolata, a fine writer/editor, I have little doubt it will be a wonderful read for the non-professional math types out there.

You can check out the table of contents here, and see if it doesn't appeal to you:

http://www.amazon.com/The-York-Times-Book-Mathematics/dp/1402793227#reader_1402793227

The one oddity is that Steven Strogatz has no entries in the volume despite the popular and renowned set of math columns he wrote for the Times back in 2010 (and additional ones since then). Kolata, James Gleick, and John Markoff are among the excellent writers well-represented. Perhaps the timing of Strogatz's 2010 pieces just missed the cut, or perhaps some element of publishing made it unfeasible for those pieces, which are included in a current volume ("The Joy of X"), to also be included in this separate volume... I don't know (or maybe there is some other simple answer that Steven, Gina, or someone else can explain in the comments below? -- at any rate, if there is ever a '2nd NY Times book of math,' I'd expect Dr. Strogatz to be represented.)
Anyway, when I can find the time, I expect to savor these pieces (many of which I've read before) covering a nice range of accessible math topics.

"read it, write it, explore it, and communicate it with confidence"

Phillips Exeter Academy

For all the secondary educators out there, I thought this goals statement on mathematics teaching, from Phillips Exeter Academy (one of the foremost private preparatory schools in the nation) was worth passing along. The link comes via a recent Twitter post from Mr. Honner.
The Academy puts the emphasis for math courses on problem solving, both doing and presenting, and is very student-centered, interactive, and collaborative. I found this explanatory paragraph especially interesting (though of course it wouldn't be practical in all secondary settings):

"As in most Academy classes, mathematics is studied seminar-style, with students and instructor seated around a large table. This pedagogy demands that students be active contributors in class each day; they are expected to ask questions, to share their results with their classmates, and to be prime movers of each day's investigations. The benefit of such participation in the students' study of mathematics is an enhanced ability to ask effective questions, to answer fellow students' inquiries, and to critically assess and present their own work. The goal is that the students, not the teacher or a textbook, be the source of mathematical knowledge."

But read the whole piece for yourself:

http://www.exeter.edu/academics/72_6532.aspx

A Little Catch-up

Just a small catch-up potpourri of things from this week's world of math. Besides the Yitang Zhang news related to prime number gaps, the other big news was Harald Helfgott's proof of the "ternary" or "weak" Goldbach conjecture (every odd number above 5 is the sum of three prime numbers). Evelyn Lamb covers it well at her Scientific American "Roots of Unity" blog here:

http://blogs.scientificamerican.com/roots-of-unity/2013/05/15/goldbach-variations/

Speaking of Scientific American, I was happy to see Joselle Kehoe, who blogs at Mathematics Rising, get to do a guest blogpost for them today (math, language, biology/evolution, and reality in a single post!):

http://tinyurl.com/atg7mbx

Mark Chu-Carroll over at "Good Math, Bad Math" blog has initiated a series of posts on "discrete probability theory." Ought to be good given Mark's methodical approach to most subjects. His first introductory post is here:

http://scientopia.org/blogs/goodmath/2013/05/12/probability-and-interpretations/

And, in the event that "probability" is your thing, here's another page of links to some sources on probability theory (based largely on the work of E.T. Jaynes):

http://bayes.wustl.edu/

Finally, if you're in the mood for a puzzle, Futility Closet has put up a little algebra brain-twister (from the 2003 Moscow Mathematical Olympiad) here:

http://www.futilitycloset.com/2013/05/16/going-up/

Weights and Standards... and Math

Lots of interesting math happening lately but it's getting covered well by other blogs/sites so I'll opt for something different today… this 12-min. video from the always interesting "Veritasium" that crosses the bounds of physics, chemistry, and math. It deals with measurement, and the roundest object in the world (and the difficulty of standardization); gets especially interesting from about the 4-min. mark on (but watch from the beginning):

The Exhilaration of Math... (Zhang/primes)

"Math is a beautiful thang!" (I can't say that often enough)… or perhaps I should say, math is a beautiful Zhang!!

The words I've seen describing the newly-announced work of mathematician Yitang Zhang include not just "beautiful," but also "breakthrough," "incredible," "astounding," "stunning."

If you somehow missed the math buzz of the last 48 hours, Zhang claims to have proven that there are an infinite number of paired primes that are some given finite distance apart… and that distance apart is less than 70,000,000… that's right, I said 70 million digits apart!!  And, mathematicians are THRILLED! Thrilled because this is the first time someone has shown there to be an upper limit or boundary to what the gap between adjacent primes can be. This may be a first, sort of baby step, toward trying to prove the long-held (but unproven) 'twin-prime' conjecture (that there are an infinite number of paired primes with a gap of 2).
Yeah, sure, working down from 70,000,000 to 2 might require some work yet, but ya gotta start somewhere (and you have to understand that prior to Zhang the only boundary was infinity, so stepping us down from infinity to 70,000,000 is no minor accomplishment -- though it can be difficult for lay folks to comprehend how teeny a number like 70,000,000 is within mathematics).

The Zhang story began on the Web (so far as I'm aware) with Peter Woit's rumor report Monday on his "Not Even Wrong" blog, that a special seminar by Zhang, was about to take place at Harvard, related to “bounded gaps between primes.”
Following the seminar, and its positive reception, the story got picked up quickly, first by Nature, here:

http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989

…and then by others, including these two pieces, also very accessible to the lay public:

http://io9.com/mathematician-makes-astounding-prime-numbers-breakthrou-505577996

http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html

If you haven't followed the story, read some of the above links to catch up.
I won't try to re-cap what those articles tell you, but will simply comment on how fun it is to see mathematicians get excited over a piece of news (and also watch the press pick up on it). One suspects most of the public will merely scratch their heads in bemusement that mathematicians can be so enamored of a number like 70,000,000 when they are really aiming for 2! I wish, alas, we could translate the excitement and wonderment into some form that the general public might better fathom! I can only hope that those who don't grasp what all the fuss is about, might at least be fascinated that human minds exist which are subject to exhilaration by such findings!

(Having said all this, I should add that while the experts seem optimistic, Zhang's proof still must pass vigorous peer review.)

ADDENDUM (5/19/13): a very good follow-up to the story from the Simons Foundation here:

http://tinyurl.com/am5fvjp

Math Contest (High School)

Steven Strogatz is urging folks to visit Math-o-vision's 10 math video finalists page to vote for their favorite entry of 2013, with significant prize money for some lucky, and creative, high school students! (note: you must log into Facebook to be able to vote) Deadline for voting is May 14th, so hop to it!

http://www.math-o-vision.com/  (and then click on "finalists" tab)

To whet your appetite, here are two examples of finalists, but check 'em all out!:

"The Fractalist" Review

Fabulous NY Times review from Jim Holt of The Fractalist (Benoit Mandelbrot's memoir) here:

http://tinyurl.com/cek5y92


...And, on an entertaining side-note, Clifford Pickover often posts "shiver in awe" tweets about mathematical factoids that are astounding to learn of. And this one just about did make me shiver in awe:

"The string 62644957128 occurs at position 62644957128 in the decimal digits of Pi. tinyurl.com/blrm9ar " WHOOOOA!!

Happy Birthday Richard!

He'd be 95 today....

Richard Feynman on math:

I Gotta Love This!

see here:

http://math-fail.com/2013/05/riemann-hypothesis-bumper-sticker.html

On investigation I discovered that these have actually been around for awhile, and are given out at American Mathematics Society meetings, but I couldn't find if they were for sale anywhere either on the AMS website or elsewhere. If someone knows about general availability to the public let us know.

"the dark unknown of mathematics"

 Is Shinichi Mochizuki "travelling alone"...?:

Hat tip to The Aperiodical for pointing to this fabulous, longread on Shinichi Mochizuki’s claimed 2012 proof of the ABC conjecture:

http://projectwordsworth.com/the-paradox-of-the-proof/

It's not a technical piece, but an overview of the issues/controversy generated by the proof (including its incomprehensibility!), and the need to resolve its accuracy.
I'd pick out several choice quotes, but it's all so good I just recommend setting aside some time to read the whole article, which does end as follows:

" [Mochizuki] may have found the key that would redefine number theory as we know it. He has, perhaps, charted a new path into the dark unknown of mathematics. But for now, his footsteps are untraceable. Wherever he is going, he seems to be travelling alone."

Juggling… Not Mere Fun & Games

"Ultimately, juggling holds an aesthetic as well as intellectual appeal for the mathematician. 'The way that I feel when I look at a nice equation is the same way I feel when I look at a nice juggling pattern,' said Burkard Polster of Australia’s Monash University, who literally wrote the book on the mathematics of juggling in 2002. 'There’s nothing superfluous there.' "

That's the last few lines nicely summing up a wonderful Jennifer Ouellette piece on the 'mathematics of juggling' over at the Simons Foundation site (includes an excellent 6-min. video, narrated by George Hart, as well as additional links). Have a look:

https://simonsfoundation.org/features/science-news/model-behavior-the-mathematics-of-juggling/

Riemann, Dyson, Montgomery, Quasi-crystals, Matrix Theory...

....and, quantum physics.

Below a longish and deep read from the Institute For Advanced Study on the mysterious linkage between the Riemann zeta function and physics (a lot to chew on or just try to comprehend):

https://www.ias.edu/about/publications/ias-letter/articles/2013-spring/primes-random-matrices


Just a couple of passages from the long piece:

 “ 'The Riemann zeta function remains one of the mysteries of modern mathematics. It is a function that we understand a lot about except for the most important question,' says Peter Sarnak, Professor in the School of Mathematics. 'It connects the theory of prime numbers, or encodes deep information about the theory of prime numbers, with the zeros. It controls the prime numbers in a way that nothing else we know does. While understanding prime numbers is an important problem, it is the generalizations of the Riemann zeta function and the objects associated with these that make it more significant.' ”....

"Quasi-crystals were discovered in 1984 and exist in spaces of one, two, or three dimensions. Dyson suggests mathematicians obtain a complete enumeration and classification of all one-dimensional quasi-crystals, the most prevalent type, with the aim of identifying one with a spectrum that corresponds to the Riemann zeta function and one that corresponds to the L-functions that resemble the Riemann zeta function. If it can be proved that a one-dimensional quasi-crystal has properties that identify it with the zeros of the Riemann zeta function, then the Riemann Hypothesis will have been proved."

I haven't had occasion to read the 2nd and 3rd volumes of Matthew Watkin's trilogy on prime numbers, but I can't help but think that the above article may overlap with or relate to some of the notions reached by Watkins who also writes of the 'vibrational' nature of the number system.

Math In the Modern World via Richard Elwes

"Chaotic Fishponds and Mirror Universes" is the new volume from perhaps my favorite, little-known-in-the-US-but-altogether-worthwhile-knowing-about math popularizer, Richard Elwes. He hails from Britain, and unfortunately his books often don't achieve wide distribution or publicity over here.
Read about the new volume here (I haven't read it yet myself):

http://richardelwes.co.uk/2013/05/03/chaotic-fishponds-and-mirror-universes/

Richard's personal webpage is here:

http://richardelwes.co.uk/

And the rest of his works, through Amazon listed here:

http://tinyurl.com/co6x678

Or you can sample some of his writing at plus.maths.org here:

http://plus.maths.org/content/list-by-author/Richard%20Elwes

Richard would easily make it onto my list of 5 favorite current math popularizers (maybe even top three!)... check him out.

Möbius Music

For the musically-inclined, the genius of Bach and Vi Hart in a single post!... ;-)

A hat tip to Steven Strogatz for leading me to this wonderful animation illustrating Bach's oft-cited Crab Canon played as a möbius strip:


[taken from here: http://strangepaths.com/canon-1-a-2/2009/01/18/en/ ]

This in turn reminded me of one of Vi Hart's early renditions with a manual music box playing a möbius strip:


[from here: http://www.youtube.com/watch?v=3iMI_uOM_fY ]

Books For Young and Older

"Children dream big. They crave exciting and beautiful adventures and they love to pretend-play."

That's how a piece over at Scientific American blogs begins on the new book from Maria Droujkova and Yelena McManaman, "Moebius Noodles: Adventurous math for the playground crowd." Children have a natural talent for math and the authors intend to exploit it (for the good of the children and their teachers), by creating "rich, multi-sensory, deeply mathematical experiences for young children... with a bit of know-how every parent and teacher can stage exciting, meaningful and beautiful early math experiences... The everyday world of children turns into a mathematical playground." If you have children, or are a teacher, definitely take a look:

http://blogs.scientificamerican.com/budding-scientist/2013/04/29/adventurous-math-for-the-playground-set/

BTW, Maria was interviewed earlier this year for a Wild About Math podcast here:

http://wildaboutmath.com/2013/03/01/maria-droujkova-inspired-by-math-24/


And on a different level I'll direct readers to my own latest interview, with Vickie Kearn, up at companion blog MathTango:

http://mathtango.blogspot.com/2013/05/vickie-kearn-she-reads-em-before-you.html

If you know who Vickie is you'll enjoy it… if you don't know who Vickie is you OUGHT get to know her… some day, you just might be in a position of wanting to speak with her… ;-)